A231809 Denominator of asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p is {K*p*(p-1)/2 : K integer}.
1, 3, 5, 5, 165, 165, 561, 561, 12903, 170085, 170085, 170085, 170085, 170085, 55957965, 141368472245, 25022219587365, 25022219587365, 25022219587365, 1776577590702915, 1776577590702915, 1776577590702915, 287890168626762845, 4749253940274679, 4749253940274679
Offset: 1
Examples
0, 1/3, 2/5, 2/5, 67/165, 67/165, 230/561, 230/561, 5317/12903, 70307/170085, 70307/170085, 70307/170085, 70307/170085, 70307/170085, 23158993/55957965, 58560723101/141368472245, 10373287618037/25022219587365
Links
- José María Grau Ribas, Table of n, a(n) for n = 1..40
- Jose María Grau, A. M. Oller-Marcen, and Jonathan Sondow, On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013-2014.
Programs
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Mathematica
<< DiscreteMath`Combinatorica` (*ver 5.0*) << Combinatorica` (*ver 8.0*) fa[n_] := FactorInteger[n]; lcm[lis_] := lcm[lis] = {aux = 1; Do[aux = LCM[aux, lis[[i]]], {i, 1, Length@lis}]; aux}[[1]]; inclusexclus[lis_] := inclusexclus[lis] =Sum[(-1)^(1 + Length[lis[[i]]])/lcm[lis[[i]]], {i, 1, Length@lis}]; densidad[lis_] := Sum[inclusexclus[KSubsets[lis, i]], {i, 1, Length[lis]}]; lista[n_] := Table[(Prime[i]^2 - Prime[i])/2, {i, 2, n}]; Table[Denominator@densidad[lista[i]], {i, 1, 15}]
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